A small confectionary store serves take away fruit drinks, operating seven days a week. The

daily operating hours is from $7\mathrm{.}00$AM until $3\mathrm{.}00$PM$.$ Each fruit drink costs $$10$ per cup. Past

data shows that on average, $9$ customers arrive at the store each hour. The store employs one

staff, and, on average, it takes $4$ minutes for the staff member to serve each customer $($from

taking the order and the payment as well as making and serving the drink$).$ The hourly salary

of the staff member is $$25.($Assume Poisson arrival and exponential service$)$

Answer the following questions:

a$)$ What is the probability that more than $3$ customers are in the store at any one time? $(1$ mark$)$

b$)$ Suppose the store manager wants to reduce the probability in sub$-$question a$)$ by $50\%,$ how

would that decision affect the service rate? Use trial and error to find the answer, and

verify your answer by checking the probability of sub$-$question a$)$ using the service rates

that are $0\mathrm{.}1$ above and below your final answer. $(1$ mark$).$

c$)$ Assume that customers do not want to wait in a queue for more than $3\mathrm{.}5$ minutes, what is

the average number of customers that the store would lose per day? $(1$ mark$)$

d$)$ Calculate the total time customers spend in the store $($from waiting until getting their

order$)$ if the store now takes two options as below:

$($i$)$ open another counter which will be served by another staff member with a similar

level of skills to the original staff to perform the full service to customers. $(\frac{1}{2}$ mark$)$

$($ii$)$ replaces the current staff member with anautomatic machine that has a service time

$($assume normal distribution$)$ of $4$ minutes. $(\frac{1}{2}$ mark$)$

e$)$ If the staff member wants to match the average total customers' time in the system that is

achieved by the automatic machine in sub$-$question d ii$),$ how quickly will the staff

member have to serve each customer? $(1$ mark$)$